Alex is the son of Roderick Burgess, the man who imprisoned Dream instead of Death. He learned magic from his father, and ultimately takes over his father's estate and magic order.
Like his father, he kept Dream imprisoned, also trying to bargain immortality for release. Dream still was stoic, and stonewalled the son as he did the father. Alex became obsessed with Dream, to the point of reading the entry of Dream in the Liber Fulvarum Paginarum over and over again. This is the only magic book he'll read and Alex eventually passes leadership of the Order on to his boyfriend, Paul McGuire, who was a gardener at the Burgess estate. Finally in 1988, Alex is a broken old man like his father. He threatens Dream to no avail. Alex's caretaker breaks the circle that holds Dream by turning Alex's wheelchair over it. Dream plays possum, pretending to be dead. When the guards come in to inspect what went on, Dream takes a handful of sand, sends them to sleep and escapes to the Dreaming. Weakened from years of imprisonment, Dream recharged himself. While Alex was taking a nap, Dream visits him. Alex pleads for his life, stating it was his father that imprisoned him, and that they wanted to imprison Death instead. Dream reminded Alex that he didn't release him, and tried just like his father to bargain for things he shouldn't have and Dream couldn't give anyways. So as an act of revenge, Dream gave Alex "eternal waking," in which he dreams that he keeps "waking up" from one nightmare to another. Alex is stuck in this nightmare for years, until finally Dream forgives him and releases him.